# Cutting a cake into $2$ pieces of equal area

You have a large rectangular cake, and someone cuts out a smaller rectangular piece from the middle of the cake at a random size, angle and position in the cake (see the picture below). Without knowing the dimensions of either rectangle, using one straight (vertical) cut, how can you cut the cake into two pieces of equal area?

Here is my attempt at the solution. I assume that all the dimensions are known.

Area of large rectangular cake $=ab\text{ }(a,b\in \mathbb{R}^+,a>b)$

Area of smaller rectangular piece $=cd\text{ }(c\in [0,a],d\in [0,b],c>d)$

Total area $=ab-cd$

Area right of vertical cut $=mb$ (we need to determine $m$)

Area left of vertical cut $=(a-m)b-cd$

Equate both expressions: $mb=(a-m)b-cd$

$$cd=(a-2m)b$$

Therefore

$$m=\frac{1}{2}\left(a-\frac{cd}{b}\right)$$

Is there a way to answer the question without assuming we know the dimensions.

• The word "vertical" in the question is confusing. You might change it to, "one straight cut". – daniel Oct 8 '16 at 20:34
• @daniel, if we imagine a 3D picture of a rectangular cake, the problem is trying to say that we should not cut the cake horizontally (half way of its height) because cutting horizontally will also solve the problem and makes two equal parts. Therefore vertical cuts in here means to cut the cake from top surface to its bottom. – Seyed Oct 8 '16 at 22:49
• @Seyed: yes i know what the problem is trying to say. a 'vertical line' with reference to the picture is not what is meant. – daniel Oct 9 '16 at 5:13
• It's probably not necessary to even use the phrase "vertical cut"... Assuming we are dealing with separating the new shape into two equal parts in terms of area, we can just deal with a single "cut" along the provided plane. – Eliseo d'Annunzio Feb 13 at 1:10